This is, or really I should say "was", a well-known property. The locus of points from which an ellipse subtends a right angle is usually called the "director circle", because the corresponding locus for a parabola is its directrix.
Here's something you may like. Take a loop of string around your` ellipse, into which you're going to put your pencil-point and use it to draw a circumscribing curve, in a way that specializes to the usual way of drawing an ellipse when the starting ellipse degenerates to a line segment.
Then the curve you draw will be a confocal ellipse to the original, and both the part of the string that touches the inner ellipse, and the part that doesn't, remain of constant length.